Method for form-cutting teeth on non-cylindrical blanks



March 28, 1961 v E. WILDHABER 2,976,773

METHOD FOR FORM-CUTTING TEETH ON NON-CYLINDRICAL BLANKS Filed March 21, 1956 4 Sheets-Sheet 1 F|G.I FIG.2 FIG?) INVENTOR. v

Ewt- \)\1 Law March 28, 1961 E. WILDHABER 2,976,773

METHOD FOR FORM-CUTTING TEETH ON NON-CYLINDRICAL BLANKS Filed March 21, 1956 4 Sheets-Sheet 2 INVENTOR.

FIG.II FIG.|2 v

March 28, 1961 E. WILDHABER 2,976,773

METHOD FOR FORM-CUTTING TEETH 0N NON-CYLINDRICAL BLANKS Filed March 21. 1956 4 Sheets$heet 3 INVENTOR.

March 19 E. WILDHABER 2,976,773

METHOD FOR FORM-CUTTING TEETH ON NON-CYLINDRICAL BLANKS Filed March 21, 1956 4 Sheets-Sheet 4 INVENTOR.

EM; wwkdg METHOD FOR FORM-CUTHNG TEETH ON NON-CYLHYDRHIAL BLANKS Ernest Wildhaber, Brighton, NIY. (124 Summit Drive, Rochester 20, N3.)

Fiied Mar. 21, 1956, $9 1. No. 573,934

17 Siaims. a. iii-4d) The present invention relates to cutting teeth that extend in a ring-shaped zone of varying diameter, such as the teeth of face couplings, of bevel gears, of hypoid gears, and of worm-gears that have larger diameters at the tooth ends than at their middle.

Teeth of this kind present great difiiculties to form cutting both members of a pair of gears or couplings having helical tooth sides, that is to describing an entire tooth surface in a single pass of a cutting edge, on each member. That is because the tooth profiles change lengthwise of the teeth. Thus on conventional bevel gears with straight teeth pointing towards the apex of the gear the tooth profile is smaller and more curved at the inner end of the teeth than at their outer end. On face couplings with helical tooth sides the tooth profile is substantially straight in the cross-section of a tooth side, but the profile inclination changes lengthwise of the teeth.

Form-cutting methods are very attractive from the production point of view, and they permitto produce particularly smooth juncture portions between the side profiles and the tooth bottoms, at the region where stresses are largest.

One object of the present invention is to devise an efficient method for form-cutting both members of a pair, even though the teeth have a varying distance from the turning axis of the respective member.

A further object is to alter the tooth surfaces, so that they permit form-cutting with tools moved across the toothed face, while also fulfilling the kinematic requirements. 7

On bevel gears for instance I may proceed in one of two ways. Either I effect a pressure angle or profile inclination that increases from the outer end to the inner end of the teeth, so that a constant profile curvature is in order. The increase in pressure angle then makes up for the decrease in diameter. Or I use a cutting edge of varying curvature, as one of involute shape, and move it so that the more curved portion of the cutting edge approaches the pitch lineas the cutting edge moves from the'outer end towards the inner end of the teeth. On face couplings with helical tooth sides I obtain a closeapproxipendicular to said plane, that is at an angle smaller than 45 degrees.

It maybe parallel or nearly parallel to'said plane.

A still other object is to form-cut the teeth of eachv member of a pairby moving a form-cutting edge in a straight path across theface of the respective member, while' also tilting the cutting edge in timed relation and in proportion to the motion'along said path.

A further aim is to form-cut the teeth of each member elicoids whose axes extend paralof a pair of spiral bevel gears, hypoid gears and some wormgears, by moving a cutting edge or cutting edges across the face of a gear blank while tilting said edge about an axis inclined to the direction of the axis of the blank, and while simultaneously turning the gear blank on its axis in timed relation to said motion across the face.

Furthermore a method shall be devised of form-cutting both members of a pair of gears having angularly disposed and offset axes, where each cutting edge of a plurality is moved in an approximately straight path across the face of a gear blank and is simultaneously'tilted about an axis inclined to the direction of the blank axis, to describe a twisted surface in space, and where also the gear blank is turned on its axis in timed relation thereto. These motions add up to a cutting motion in which a cutting edge completely describes a tooth side in a single pass across the face.

This form-cutting process for both members of gear pairs having angularly disposed and offset axes requires novel tooth shapes. These have been described at length in my pending application entitled Gearing, filed November 1st, 1955, Serial #544,270, Patent No. 2,930,248 issued March 29, 1960; and reference is made to it. In one aspect the present application represents one method of producing the tooth shape there described.

Another method to this end is described in my pending application entitled Method and Means for Cutting Spiral Teeth, filed January 3, 1956, Serial #557,151, to which reference is made. This method is related to the method of the present invention, and refers to form-cutting without the above-named tool-tilting motion. This tilting motion during cutting is characteristic of the present invention.

Other objects will appear in the course of the specification and in the recital of the appended claims.

In the drawings:

Fig. 1 is a fragmentary end view of a tooth space of a face coupling, looking along one of its helical tooth sides towards the coupling axis,.and showing a tool in engagement therewith. It illustrates one application of the present invention.

Fig. 2 is a View similar to Fig. ,1, with a roughing tool in-place of the finishing tool of Fig. 1.

Fig. 3 is a view similar to Fig. 2, showing a modified roughing tool.

. Fig. 4 is an axial section of a coupling member having straight radial teeth with helical tooth sides, shown with a roughing tool and a finishing tool at the start of a cutting stroke; the section being taken along lines 44 of Fig. 5.

Fig. 5 is a bottom view corresponding to Fig. 4, and taken along the coupling axis.

Fig. 6 is a front view, taken along its axis, of a roughingv tool also shown in Figures 4 and 5.

Y'Figures 7 to 13 illustrate one application of the inven tion to cutting bevel-gear teeth, particularly straight teeth.

Fig. 7 is a cross-section through a tooth space of a bevel gear whose teeth have approximately constant depth from end to end, looking towards the small end of the teeth. It also shows a tool in two different cutting positions. The section is taken at right angles to the lengthwise direction of one tooth side, at its middle.

Fig.8 is a cross-section through a tooth space of a bevel gear whose teeth have taperingdepth, with a tool shown in two different positions. angles to the axis of tool tilt.

Fig. 9 is a fragmentary axial section of a bevel gear,

taken along lines 9-}9 of Fig. 8, showing also a cutting 7 Fig. 8, showing a modi- The section is taken at rightv Figures 11 and 12 are fragmentary sections taken along the pitch cone of a bevel gear, and developed into a plane, showing two different tool arrangements.

Fig. 13 is partly a side View, partly an axial section, of a bevel gear such as may be produced with the method and means illustrated in Figures 7 to 12.

Figures 14 to 18 illustrate a modified application of the invention to cutting the teeth of tapered gears, particular- 1y straight teeth of bevel gears.

Fig. 14 is a cross-section of teeth of a bevel gear, shown in engagement with'a pair of tools. The section is taken along lines 14-14 of Fig. 15.

Fig. 15 is a section along lines 1515 of Fig. 14, taken through and containing the bevel gear axis.

Fig. 16 is a cross-section similar to Fig. 14, shown in engagement with a tool. It corresponds to a larger and substantial gear reduction, in which the larger gear of the pair may be provided with straight profiles, as indicated.

Fig. 17 is across-section through the teeth of a pinion similar to the pinion of Fig. 16, illustrating one way of obtaining eased-off tooth ends.

Fig. 18 is a diagrammatic plan view corresponding to Fig. 17, to define the directions referred to.

Figures 19, and 21 are corresponding pitch surface views and diagrams explanatory of an embodiment of the present invention in which a gear blank turns on its axis in timed relation to the tool motion, and referring to gear pairs with angularly disposed and offset axes. Fig. 19 is a view at right angles to the axes of both members of the pair. Fig. 20 is a view in the direction of the axis of the gear member. Fig. 21 is a view taken at right angles to the line of contact of the pitch surfaces, along a plane parallel to the axes of both members.

Figures 1 to 6 illustrate an application of the present invention to cutting face couplings, whose straight radial teeth contain helical tooth sides. The conventional way of shaping such teeth with reciprocatory tools produces a passable surface finish only by using a great many cuts, in finishing a tooth side. A tool then applies the finishing cut with its rounded corner located at the end of the cutting edge. This finishing portion is of necessity sharp 1y curved, whereas the profile to be produced is practically straight.

With my invention an improved finish is attained with fewer cuts. A single cut applies the final finish on an entire tooth side.

The helical tooth side to be produced can be considered described by a radial line that is moved along the coupling axis and is simultaneously turned about said axis in direct proportion to the axial displacement. Fig. 1 is a view along such a radial line (35), which projects into a point, as it is perpendicular to the plane of the drawing. This line interesects the axis 36 of the helical tooth side 37. The outside profile 33 of surface 37, and its inside profile 40 are helices of the same lead, appearing in this view as practically straight lines of different inclination to the direction of axis 36. As known, the trigonometric tangent of the inclination angle is proportional to the helix' diameter. Like the helices themselves, the crosssections taken at right angles to the radius 35 of side surface 37 are also practically straight, and practically coincide with the helices.

Accordingly I may form-cut such tooth sides by moving a tool 41 lengthwise of the tooth side 37, while also tilting the tool to conform with the varying inclination of the tooth profile. The straight cutting edge 42 of the tool then moves along radial line 35. This cutting edge coincides with line 413 when it passes through the inside end of the teeth, and witlrline 38 when it passes through the outside end. 7 g e v 7 V In principlethe tool 41 may be tilted in its stroke on an axis coinciding with radial line 35. This changes the tooth depth slightly along the length-of the teeth, whereas a constant depth ispreferredon these couplings. It may be accomplished by tilting the tool 41 on an axis 43 'p'ar-.

allel to radial line 35 and parallel to the lengthwise direction of the engaged tooth side 37. Axis 43 intersects the mean normal 44 of the tooth side, and lies directly above the point 45 where the mean tooth profile (not shown) intersects the tooth bottom. It can be shown that also a slight and generally desirable amount of ease-off is produced at the tooth ends when the tilt axis is at 43.

The inclination of the tooth profile to the direction of the coupling axis is not exactly proportional to its radial distance from the coupling axis, as it is defined by the aforesaid tangent relationship. Accordingly the most exact tooth sides are attained by tilting the tool at a rate not exactly proportional to the tool travel, and determined from the trigonometric tangent of the inclination angle.

7 cutting tool 41v is the tool already described with Fig. 1.

, 56' is the first one to cut.

. workpiece.

Side-cutting tool 46 operates on the opposite tooth sides, which are helical surfaces of opposite hand. Accordingly the two tools 41 and 46 have to describe helical surfaces of opposite hand, and move helically along arrows 41, 46. Tool 41 is tilted in clockwise direction on its axis 43, viewed from the outside, as it moves inwardly. Tool 46 is tilted in counter-clockwise direction on its axis 47, as it moves inwardly. This pair of tools contains the finish-cutting edges for opposite sides of the teeth.

For roughing, other kinds of tools may be used, like tools 50, 51. These are also reciprocated lengthwise of the coupling teeth, but do not need to be tilted. Thus tool 50 moves a distance 5253 (Fig. 4) during the cutting stroke.

Preferably the work piece 49 is indexed after every cutting stroke, during the return stroke, while clear of the tools. To clear, it is withdrawn along its axis 36 a distance exceeding the tooth depth, after every cutting stroke, and advanced again into cutting position immediately before the cutting strokes.

Although the tools 50, 51 may look like gear-shaper cutters, they are not generating tools. Their outside edges 54 (Fig. 6) are at a constant distance from the tool axis 55, but having a varying width. The widest cutting tooth The narrowest cutting tooth 56" is the last one. The cutting teeth 56 act somewhat like the tools of a turret lathe. The tool (50 or 51) is indexed fromtime to time, preferably during a return stroke, to present a more suitable cutting tooth to the The inclination of the tool axis 55 (Fig. 4) is for obtaining, cutting clearance with cutting teeth that have a constant distance from the tool axis. Such a design lowers the tool cost. The cutting stroke is in a direction 52--53 along a tooth side of the workpiece, and inclined to the tool axis 55. Y

the enlarged view, Fig. 2. This .figure shows a tooth space in a radial view along one tooth side (37)., like Fig. 1. The profile 60 cut by this tool approximates the contour seen in this view, and has a constant small distance therefrom. It has an obtuse corner 60' directly opposite the radial line 35. Such a profile gives the best I approximation of tooth side 37 in a straight cut without tilt. x y The lines 61 represent the outside cutting edges of the successive cutting teeth 56 of tool 51. Edge 61' cuts first, edge 61" last, to full depth. While these edges are shown straight, they may also be circular arcs concentrio with the tool axis.

The depth feed is preferably made in small steps, each made during a return stroke, as during each return stroke during roughing. The lines 61 represent the last cutting position of the respective edges.- After this relative position is reached, the tools 50, 51 are indexed, preferably during the following return stroke, to move the next cuting tooth into operative position.

The side-cutting tools 41, 46 start to cut only after most of the stock has been removed, and are thus saved for the finishing cut. Only about in the depthwise position indicated in dotted lines 41" do these tools reach the contour cut by the respective roughing tools 51, 50, and start cutting.

Fig. 3 shows roughing teeth of modified shape. Here the contour 62 of the space is cut straight, and has no break or corner. The cut is then taken in a direction that is radial at point 63, at the tooth bottom. The projection of the coupling axis 36 then passes through point 63. The lines 64 represent the end cutting edges of successive roughing teeth, and correspond to lines 61 of Fig. 2.

While here a somewhat simpler shape of the roughing cutters is achieved, this roughing cut leaves more stock to be removed by the finishing tools.

The described process gives an efficient stock removal and a smooth finish, that is applied on each tooth side in a single pass, at full-depth position.

Cutting bevel gears Figures 7 to 13 illustrate one Way of applying my formcutting process to bevel gears, especially bevel gears having straight teeth. These teeth point towards an apex that coincides with the intersection point of the axes of a bevel-gear pair. If x denotes the distance (080, Fig. 9) of a considered point (80) of the pitch surface from said apex (0), and p denotes the profile inclination or pressure angle at said point, then the curvature radius of the cross-sectional profile is known to be proportional to xsinp On conventional straight-tooth bevel gears the pres sure angle p is constant along the length of the teeth. And the curvature radius is then proportional to x.

In accordance with one aspect of the invention, the pressure angle is changed along the teeth just so much that a constant profile curvature results. Then a: sin p=constant=C; sin 1) This is illustrated in Fig. 7, which is a mid-section taken at right angles to the pitch element 66, or to the mean straight-line element, of a tooth side 67,, and a view of its tooth space. Element 66 projects into a point, as it is at right angles to the drawing plane. Fig. 7 also shows a tool 68 with concavely curved side-cutting edge 70. The cutting edge shown in full lines is in its posi tion at the mid-section. Dotted lines 68', 70' show the tool and cutting edge at the inner end of the tooth. 71 is the projected axis of the bevel gear. It contains the apex (0) at the intersection with said element.

The cutting edge is also the gear profile. Edge 70 has a curvature center 72 lying on the tooth normal 73. Edge 76 may be either a curve of varying curvature or' a circular are. 72. is then the center of this are. p denotes the pressure angle at the midpoint, and the inclination of normal 73.

Cutting edge 76' of tool 68 is tilted or turned with respect to edge 7% about an axis close to element 66 and parallel thereto. 74 is such an axis, so positioned that'an approximately constant tooth depth resultsalong the 6 length of the teeth. Point at the end of the cut-ting edge moves peripherally to 75'.

As the tool 68 is turned about axis 74 it is also moved along it. When the cutting edge has reached its position 70' at the inner end of the engaged tooth side 67, its curvature center has moved from 72 to a position 72. The connecting line 72-66 is the new normal. Its inclination p should fulfill the above equation, using the x=value for the inner end of the teeth.

The resulting pressure angle change (p-p is not exactly proportional to the distance travelled along axis 74, nor is the turning angle about said axis. However the variation is slight, and good results are obtained by using a constant proportion of turning motion and translation. in this case the tooth side described by the cutting edge is a helical surface of constant lead, or briefly a helical surface or helicoid.

The conventional bevel gears have teeth of tapering depth, a depth larger at the outer end of the teeth than at their inner end. Figures 8 and 9 illustrate a way of obtaining tapering tooth depth with my form-cutting process. Fig. 8 is a mid-section of a tooth side 76 and a view in the direction of the axis (83) about and along which the cutting edge 77 of tool 78 is moved. For convenience let it be assumed that edge 77 lies in the plane of the drawing, which is perpendicular to said axis 83. The curvature center 82 of profile 77 lies on tooth normal 79, and axis 83 also passes through said normal. Axis 83 is offset from profile 77 and its means point 80 in a direction towards the curvature center 82.

The tool 73 moves along and about axis 83 to or from the innerend of the engaged tooth. Its position there is shown 'in dotted lines '78, its cutting edge being at 77. 82' is the new position of the curvature center. While in the middle position the cutting edge 77 moves in the direction of the tangent plane '81 at means point 89, it projects increasingly beyond said tangent plane with increasing distance from the mid-position. For center 82 moves in an are 8282- that is concave towards the tangent plane. The surface described by the cutting edge intersects the tangent plane in a line inclined to the direction of axis 83. This line moreover is approximately straight when profile 77 is circular and the turning angle about axis 83 is proportional to the displacement along said axis. That is, in this case the intersection line 84 (Fig. 9) of the tooth surface with the tangent plane can be shown to have zero curvature at mean point 80. The inclination j of line 84 to the direction of axis 83 appears in projection in Fig. 9. Tan j can be shown to amount to where Line 84 lies in the tangent plane at 8% to the tooth surface. This plane is perpendicular to the drawing plane ofFig.8.

When teeth without ease-off at the tooth ends are desired, the pitch element of the tooth surface should extend along line'84. In the case of unequal tooth addenda on the two members of a bevel gear pair, the

mean element of a tooth surfaceshould coincide with line 84.

Oneway of attaining ease-off at the tooth ends is to let the pitch element or mean element of the tooth surfaces go in a direction more inclined to the direction of axis 83 than line 84. Thus it may extend along a line connecting the points Sttand Pointed is beyond 7 point 85 of'line 84, and at a small distance from the tangent plane 81. i

Further ways of attaining ease-01f and of attaining tapering tooth depth are feasible. One of these has been described in my application Serial #544,270.

Fig. '10 is a section and view similar to Fig. 8 and showing the same tool 78. But here the axis 83', along and about which the tool 78 is moved, does not intersect normal 79 but is offset from it. The same tangent plane 81 at point 80 is attainable, as in Fig. 8, by using a tilted axis 83', rather than one that is perpendicular to the drawing plane. The amount of tilt required to maintain the tooth-tangent plane 8 1 at right angles to the drawing plane can be determined with the known procedures of mathematics.

Figures 11 and 12 are developments of conical sections coaxial with a gear and extending along the pitch cone, or along the mean elements of the teeth 89'. On gears with even tooth numbers, pairs of tools 87, 88; 87', 88' may be arranged to cut opposite tooth sides of adjacent tooth spaces 96, 90' (Fig. 11). Should the tools crowd each other with this disposition, one or more tooth spaces may be skipped between adjacent tools. Such arrangements are possible when the work piece is indexed after every cutting stroke.

Fig. 12 shows a disposition especially suitable on tooth numbers divisible by four. Here depth-roughing tools 91 alternate with the side-cutting tools 87, 88. The roughing tools $1 are not tilted during the cutting stroke. Other combinations may also be used.

The whole gear periphery may be filled with tools.

But it is also possible to use a single pair of tools, and to index the work piece either after every cutting stroke, or after completing the engaged tooth sides.

Fig. 13 shows a bevel gear 92 out by the process just described with Figures 8 to 12. It has a conical pitch surface 93 and an axis 94. The sides 95 of its teeth 6 are twisted surfaces, whose profile inclination or pressure angle increases from the outer end 2 7 to the inner end 98 of the teeth. This varying inclination is pointed out by the changed direction of the pro-file tangents 100, 101 at the mean profile points.

. Opposite sides 95 of each tooth are symmetrical with respect to the central plane of the tooth.

Modification 115' at the mean profile points 116, 116' happen to be in line with each other. They are tangent to an are 117 at a point 118. With conical-involute tooth sides 120, 120' the are 117 lies on the base cone: of the tooth surfaces. 0-118 (Fig. 15) is a cone element thereof. The

central pitch-cone element is indicated in dot-and-dash e lines 121.

The profiles of the tooth sides 120, 120' are exact or approximate involutes centered at 12.2. The cutting edges 123, 124 of the tools 113, 114 match these profiles.

According to the invention each tool is made to performa helical motion of constant lead'ahout an axis 125 passing through 127. and parallel to element 0-118, that is perpendicular to the drawing plane of Fig. 14. In this helical motion each cutting edge 123, 124 describes an involute helicoid. This surface is known to contain straight-line elements in. all planes tangent to its' base cylinder. The horizontal plane containing points 116,

, 118 and apex t is one such tangent plane. It is also a tangent plane of thebase cone of the bevel gear. This plane intersects the desired tooth side 120 in a straight line that'connects point 116 with apex t). Itincludes a o of decreased taper.

known angle e (not shown) with element 0-118,whosetrigonometric tangent is tan 116118 The lead of the helical tool motion is so determined in known manner, that this angle e is the helix angle at the base cylinder of the involute helicoid. Then the straight line t 116 is also a straight-line element of the described involute helicoid.

In other words, we substitute an involute helicoid for the conical-involute tooth surface, contacting therewith along the pitch element or mean element 0116. 116 118 is the required radius of curvature of the tooth profile in the mid-section shown in Fig. 14. It is also the radius of curvature of the involute helicoid described by the cutting edge 123. The curvature of the two surfaces matches exactly.

And they match exactly not merely in the mid-section. They match exactly at all points of element il116. For the curvature radius in sections parallel to the drawing plane of Fig. 14 is the distance from element tl118 of any point of element 0116, on both the conicalinvolute tooth surface and on the involute helicoid substituted therefor.

The involute helical tooth sides are therefore suitable surfaces for the tooth sides of the straight bevel-gear teeth 112.

Opposite tooth sides 12%, are helicoids of opposite hand, one being right hand and the other being left hand. In the instance illustrated two opposite tooth surfaces have the same axis 125, which lies in the plane of symmetry or mid-plane of tooth 112'. Such a disposition resulting in coaxial helical surfaces of opposite hand is however no compulsory requirement.

The same curvature radii and curvature distribution along the teeth is obtainable also with other positions of the axis of the helical tooth surface. Any axis parallel to accomplishes this when it lies in the axial plane 1 18-0122,- the drawing plane of Fig. 15. One such axis may pass through point 126 (Fig. 14), at a smaller distance from point 118 than axis 125. Another such axis passes through point 127. While such shifts produce the same profile curvatures, they affect and control the lengthwise taper of the tooth depth.

When moved about and along axis 125 to the inner end 128 of the teeth, the tools 113, 114 move to positions 113', 114' shown in dotted lines. The point 130 at the tooth bottom moves to 130' peripherally about axis 125. This direction is indicated by tangent 131, which in this instance passes through apex 0. Tangent 131 represents the general direction of the produced tooth bottom, in this view.

A tool moved about and along an axis passing through point126 has its point 139 displaced at an increased angle to the horizontal direction, and produces a tooth depth of increased taper. A tool moved about and along an axis passing through point 127 produces a tooth depth By shifting the axis of the helical motion the taper of the tooth depth may be controlled.

To change the profile curvature of the tooth profile 120, an axis may be used that passes through a point oliset from the central axial plane 118-6122, together with a dififerently curved cutting edge. To increase the profile curvature, that is to reduce the radius of curvature, the ems of the helical motion may pass through a point, such as 135, at the left of central plane 118ll-1 22. The profile of the surface described by the cutting edge should then be an involute whose base circle is centered at 135. Its curvature center is the projection of point 135 to line 116118.

The required-change of profile curvature lengthwiseof the teeth is attained exactly,- if the axis 136 passing through. point135 is not parallel to but inclined to the 9 central plane 118-0-122, so as to passthrough a-point 137 of said plane. Point 137 is directly underneath apex 0.

To decrease the profile curvature and increase its radius of curvature, the axis of the helical motion should intersect the drawing plane at the right of central plane 118-0-122.

Thus far the considered cutting edges 123, 124 were located in the drawing plane of Fig. 14 and were involutes. However the cutting edges may also be at an angle to these involutes, as long as they lie in the involute helicoid to be produced. This generalization is applicable to all embodiments described. The cutting edge does not need to be in a plane perpendicular to the axis of swing. The invention is not confined to what are called involute or octoid tooth profiles. Other profiles are also feasible. Fig. 16 is a mid-section through the contacting teeth of a pair of bevel gears 140, 141 whose larger member 14-1 contains straight tooth profiles 142. The side surfaces of its straight teeth 143 are then planes, and lend themselves also to many known production processes. The side surfaces of the mating teeth 144 then contain profiles 145 more curved than the conventional involute. 146-147 is the radius of profile curvature at mean point 146. it can be computed with the known means of kinematics. It can also be experimentally determined by generation.

The profiles 145 are curves of varying curvature, similar to an involute. I substitute an involute'therefor, that has the same radius of curvature at point 146. Any involute fulfills this requirement, if its center 148 lies on a line 150 that is perpendicular to normal 146-147.

The exact profile required can be determined experimentally or by analyzing the surface of action of the pair of gears. This permits to find the best location of center 148 on line 150.

A tooth side with profile 145 is described by moving tool 151 along and about an axis perpendicular to the sectional plane and passing through point 143. The sidecutting edge of tool 151 then describes an involute helicoid, whose lead should be so determined that the straight-line element projected into line 146-147 passes through the apex of the gear pair.

Slight departures from fully conjugate tooth surfaces are often desired, to ease off the boundaries of the tooth surfaces. Profile ease-off is attained simply by using cutting edges slightly more concave than required for full conjugacy. Ease-off at the tooth ends may be achieved in known manner when both members of a gear pair have convex profiles. It may be done by arranging the contacting straight-line elements of the tooth surfaces at a slight angle to each other, rather than letting them coincide.

How ease-oif at the tooth ends can also be attained by modifying the cutting process on only one member, or on both members, will now be described with Figures 17 and 18. Fig. 17 is a sectional view of teeth 144, showing also a tool 151' similar to tool 151 of Fig. 16. The sectional plane 160 (Fig. 18) is perpendicular to the plane 161 that contains the axis of the gear and the curvature center 162.

It can be demonstrated that ease-01f at the tooth ends can be attained by moving the tool 151' helically about an axis 163 that intersects the drawing plane of Fig. 17 at a point 164. Point 164 is offset from plane 161 in a directionaway from tool 151. To attain the required change of profile curvature along the length of the teeth, the axis 163 should be inclined in such a way that it intersects plane 161 at a point directly underneath apextl, :see Fig. 18. At the same time the-direction of the pitch'line is preferably somewhat altered by raising the apex 0, as shown in Fig. 17. p 1T have now described two distinctly different ways of form-cutting tapered gears, and especially bevel gears with straight teeth, both using a cuttingmotion about it) and along an axis. in one the pressure angle-is changed along the teeth in such a way that the profile curvature may stay approximately constant along the length of the teeth. In the other no change in pressure angle lengthwise of the teeth is called for, the effect being attained with cutting edges of varying curvature.

The two procedures can also be combined, if desired.

Spiral teeth Straight teeth are produced in accordance with the invention by moving a cutting edge in an approximately straight path across the face of a gear blank, while also tilting said cutting edge. The cutting edge then describes a twisted surface which is often a helical surface of constant lead.

To produce spiral teeth the same or similar surfaces are described in space by a cutting" edge. In addition the work piece is turned on its axis in timed relation to the motion of said cutting edge. The cutting edge may have the same shape as with straight teeth, but a different cutting face adapted to the different cutting direction. Or it may be differently positioned, and may lie for instance in a plane normal to the tooth direction at the middle portion of the teeth, or lie in any other way on the tooth surface to be described.

To cut spiral-bevel gear pairs with intersecting axes, the tool motion preferably follows the line of contact of the conical pitch surfaces of the gear pair.

More problems exist on gear pairs having offset axes .angularly disposed to each other. These will now be described further, also because they represent the general case, and conclusions reached for the general case also apply to specific cases. They apply also to spiral bevel gears with intersecting axes, where the pitch surfaces are in each instance a pair of well-defined conical surfaces that have pure rolling contact.

When however the angularly disposed axes are offset, there are no such pitch surfaces with pure rolling contact and having the same properties. The pitch surfaces then should be redefined. According to a broad definition they are surfaces of revolution coaxial with the axes of a gear pair and contacting along a line. To give them a practical meaning the teeth should extend along the pitch surfaces. Restrictions are imposed upon the lines in the pitch surfaces along which the tooth sides extend, by requiring that these pitch lines extend in the direction of relative sliding at their point of contact, that is at their spectively. The hyperboloidal pitch 'surfaces'3ti2, 3% are coaxial therewith, and contact each other along a straightline element 304 that intersects the center line 305 at a point 396. The center line 3115 is understood to be the shortest connecting line between the two axes 300, 3G1. It intersects them at points 307, 308, and is perpendicular to both axes. Straight-line element 304 is inclined to the direction of the pinion axis 300 at an angle i.

Let E denote the shaft offset 307-308, and E the offset 306-307 of line 304 from the pinion axis 300. To achieve contact of the pitch surfaces along line 304,

E, and i have to be related as angle is a right angle:

. E,.= /zE (1-cos 2i) E sin i 4 The hyperboloidal pitch surfaces are obtainable by rotating line 304 about the pinion axis 300 and about the gear axis3tl1 respectively.

follows, when the shaft Different parts of the samekind of pitch surfaces may i 11 be used on wormgears, such as the part of pitch surface 302 bounded by the two lines 302, and the part of pitch surface 303 bounded by the two lines 303'. Hypoid gears use the tapered parts of the pitch surfaces.

Assumption of the pitch surfaces determines the pitch lines 310, 311 of the pinion and gear, along which the tooth surfaces extend. The pitch lines 310, 311 contact at their intersection points with element 304. 312, 312 are such points of contact. The pitch lines contact there because they were determined so as to extend in the direction of relative sliding at these and all other points of element 304.

The inclination of the pitch lines to contact element 304 varies along said element. Fig. 21 shows the normals 314', 314" of the contacting pitch lines 310, 311, and also the normal 314 at mean pitch point 312.. These normals lie in the drawing plane of Fig. 21 and are perpendicular to the common tangents of the contacting spiral pitch lines. It can and has been shown that these normals all intersect at a common point 315 on the center line 305, and that the distance B =306-315 can be mathematically expressed as B (E E,) cos i-l-mE, sin e r m sin icos i Herein denotes the ratio of the tooth numbers N and n of the gear and pinion respectively, whose otfset axes are at right angles to each other.

The pitch lines 310, 311 are known to be uniformmotion spirals. They can be considered described on the respective pitch surfaces by a point moving at a constant rate along contact element 304, while the pinion and gear with their pitch surfaces turn uniformly on their axes at the inverse ratio in of their numbers of teeth. The rate of travel required for the describing point may be determined from the known inclination of the pitch spirals.

A cutting edge may be moved uniformly along the contact element 3% and then describes a side surface of a spiral tooth on each of the uniformly rotating gear members. Its intersection point with element 304 describes the pitch lines. It describes the same pitch lines and different but match ng profile inclinations on the two members, when the tool is simultaneously tilted about an axis passing through the describing point. In some cases this axis of tilt coincides with the contact element 304.

Various small departures may be made to achieve a tapering tooth depth, and to achieve ease-off at the tooth ends, in this and all embodiments. Procedures to this end have already been described here for bevel gears with intersecting axes. Other procedures for hyp-oid gears have been described in application Serial No. 544,270. Combinations of both may also be used. .No further disclosure is believed required regarding these and other small departures to this end. a

The tool-tilting motion has a different effect on hypoid gears and worm gears than on bevel gears with intersecting axes. It affects thedirection and intimacy of tooth contact.

in the last named application a special tooth shape is disclosed for hypoid gears with relatively large shaft offset and wormgears. It rests on the concept'of basic members, that have the same kind of instantaneous relative motion with respect to each gear of apair, as said gears have relatively to each other. Such basic -members'contact the gear teeth along the same lines as the gears contact each other. Thus arack is a simple basic member for gears running on parallel-axes. A crown gear is" a basic member for bevel gear pairs with intersecting axes.

Gears with angularly disposed and offset axes also have basic members. They are specific helical segments. An infinite number of basic members exist for any given gear pair. Its axis always intersects the center line (305). The angle i of this axis with respect to the direction of the pinion axis (300) may be assumed at will. Then the olfset E of this axis from the pinion axis, for gear pairs with right shaft angles, is determined by the formula This is exactly the same formula as given for the offset E, of pitch element 304.

The lead L of the basic member amounts to L =1r E sin 2i=21r E sin i cos i The turning ratio of the basic member can be determined by vectorial addition, keeping the direction of the instantaneous axis the same as for the gear pair. In other words, the turning ratio can be determined as if for bevel gears with intersecting axes, as if all the axes would intersect in a common point.

The above-named application also contains equations for gear pairs with shaft angles other than right angles. Reference is made to it for these less common cases.

The special tooth shape is obtained by selecting a suitable basic member and making its tooth shape a line only, that extends from top to bottom of the teeth of the pair. This line is embodied as a cutting edge. In the mesh between the basic member and each gear of the pair this line moves lengthwise of the gear teeth because of the relative lengthwise sliding present with gears having angularly disposed and offset axes. It describes on the two gears of the pair tooth surfaces that contact along the line itself. Accordingly the tooth surfaces. of each gear of the pair can be described and cut by a cutting edge moving helically about the axis of a basic member, While the gear turns on its axis in direct proportion to said helical motion. Contact between mating gear teeth is along a line which sweeps the tooth surfaces and which represents the successive positions of said cutting edge.

When the cutting edges are curved, the two gears of a pair should be produced with counterpart cutting edges.

The axis of the selected basic member may pass at a distance from the tooth-zone portion engaged by the cutting edge, or it may pass right through the engaged tooth zone or near it. It mayslie in the mean tangent plane of the engaged tooth zone; and it may be the contact element (304) of a pair of hyperboloidal pitch surfaces.

n1 any case the axis of the helical tool stroke is inclined to a plane tangent to the tooth zone engaged by the tool at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, that is at an angle smaller than 45 degrees. It may also be parallel to said plane. Furthermore it is inclined to the axial direction of the work piece.

A cutting edge describes a helical surface in space. This surface differs here from the tooth surfaces because of the simultaneous rotation of the gear blank.

I Hypoid gears with relatively small shaft olfset are pre erably cut with less of a tool-tilting motion than obtained with basic members. The cutting profiles used on the from the spirit of the invention.

This application is intended to cover any variations, uses, or adaptations of theinvention following, in general, the'principles of the invention and including such departures from the present disclosure as come Within the known' or customary practice in the art to .which the invention pertains and as may be applied to the essential '13 features herein set forth and as fall within the scope of the invention or the limits of the appended claims.

I claim:

1. The method of cutting a tooth side of a member whose teeth extend in a ring-shaped zone of varying diameter, which comprises moving a tool across the face of said member while simultaneously tilting said tool on an axis that is inclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, and thereby describing said entire tooth side in a single traverse across the face.

2. The method of cutting a tooth side of a member whose teeth extend in aring-shaped zone of varying diameter, which comprises moving a tool across the face of said member while simultaneously tilting said tool on an axis that is inclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, to effect cutting motion, said two simultaneous motions being timed with each other, effecting feeding motion between said tool and member to approach said tool and member towards one another, and repeating said cutting motion in difierent feed positions, to describe said entire tooth side in a single feed position.

3. The method of cutting a tooth side of a member whose teeth extend in a ring-shaped zone of varying diameter, which comprises moving the cutting edge of a tool across the face of said member while simultaneously tilting said tool on an axis extending approximately in the direction of the engaged tooth zone, to effect cutting motion, said cutting edge having a side portion to cut the Working part of said tooth side and a convexly curved end portion, said axis being so positioned that the points of said side portion have a varying distance, respectively, therefrom, said two simultaneous motions being timed with each other, effecting feeding motion between said tool and member to approach said tool and member towards one another, and repeating said cutting motion in different feed positions, to describe said entire tooth side in a single final feed position.

4. The method of cutting a tooth side of a member whose teeth extend in a ring-shaped zone of varying diameter, which comprises moving a tool in an approximately straight path across the face of said member while simultaneously tilting said tool about an axis that is inclined to a plane tangent to the engaged tooth zone portion at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, and thereby describing said entire tooth side in a single traverse across the face.

5. The method of cutting a face coupling member containing radial teeth, which comprises positioning a pair of straight cutting edges adjacent opposite sides of said teeth, moving said cutting edges simultaneously across the face of said member, said motion being along and about an axis on each cutting edge of said pair, so that said cutting edge describes a helical surface at least approximately, said axis extending in the lengthwise direction of the engaged tooth side, the surfaces described by said two cutting edges being of opposite hand, effecting depthwise feeding motion between said cutting edges and said member axially of said member, and repeating said helical motions in different feed positions, to describe and cut each entire tooth side in a single feed position.

6. The method of cutting straight teeth on a bevel gear, which comprises positioning a pair of form-cutting edges adjacent opposite sides of said teeth, moving said cutting edges simultaneously across the face of said gear, said motion-being about and along an axis on each of said cutting edges so that said cutting edge describes a helical surface at least approximately, said axis being nclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to 14 the direction perpendicular to said plane, the surfaces described by said two cutting edges being of opposite hand, effecting depthwise feeding motion between said cutting edges and said gear, and repeating said helical motions in different feed positions, to describe and out each entire tooth side in a single feed position.

7. The method of cutting teeth whose two tooth ends have different distances from the axis of a member, which comprises moving a plurality of form-cutting edges simultaneously in different paths across the face of said member while simultaneously tilting each of said cutting edges about an axis, to effect cutting strokes, said paths being approximately straight and said axis being inclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, repeating said cutting strokes while indexing said member after every stroke, and effecting depthwise feeding motion between said cutting edges and said member to finally describe and cut each entire tooth side in a single feed position.

8. The method of cutting teeth of changing depth on a tapered gear, which comprises moving a concave formcutting edge in an approximately straight path across the face of a gear blank while simultaneously tilting said cutting edge about an axis inclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, said axis being oifset from the cutting edge in a direction towards the curvature center of said cutting edge, said two simultaneous motions being timed with each other and constituting the cutting motion, effecting depthwise feeding motion between said cutting edge and gear blank, and repeating said cutting motion in different feed positions, to describe and cut the entire engaged tooth side in a single feed position.

9. The method of cutting teeth of changing depth on a tapered gear, which comprises moving a concave cutting edge across the face of a gear blank to describe and cut part of an involute helicoid thereon, said edge being,

moved along and about the axis of said helicoid, said axis coming close to and being angularly disposed to the axis of said gear blank.

10. The method of cutting -a tooth side of a gear whose teeth extend in a ring-shaped zone of varying diameter, which comprises moving a cutting edge across the face of a gear blank while simultaneously tilting said tool on an axis that is inclined to a plane tangent to the engaged tooth zone at an angle smaller than the angle at which said axis is inclined to the direction perpendicular to said plane, so that said cutting edge describes a twisted surface in space, turning said gear blank on its axis in direct proportion to the motion of said edge across the face, all said motions combining to effect a cutting pass, and in thereby describing said entire tooth side in a single cutting pass.

11. The method of cutting a tooth side of a gear whose teeth extend in a ring-shaped Zone of varying diameter, which comprises moving a cutting edge in a helical path across the face of a gear blank while simultaneously turning the gear blank on its axis in timed relation to the motion of said cutting edge, all said motions combining to effect a cutting pass, the axis of said helical path being angularly disposed to the direction of the blank axis, and in thereby describing said entire tooth side in a single cutting pass. I

12. The method of cutting the teeth of both members of a pair of gears having angularly disposed axes, which comprises cutting each member of said pair by moving each of a plurality of cutting edges in an approximately straight path across the face of the respective member while simultaneously tilting said cutting'edge on an axis inclinedto the direction of the axis of said member, so

thatsaid cutting edge describes a twisted surface in space, simultaneously turning said member on its axis in pro 15 portion to the displacement of said cutting edge along said path, all said motions combining to eifect a cutting pass, and thereby describing and cutting the entire engaged tooth side in a single cutting pass.

13. The method of cutting teeth having a varying distance from the axis of a toothed member, which comprises rotating said member continuously and uniformly on its axis, moving said form-cutting edge about and along another axis in repetitive strokes in time with the turning motion of said member, to describe a helical path across the face of said member, each stroke occupying a turning angle of said member contained in a full turn thereof an integral number of times, said integral number being prime to the tooth number of said member, so that said cutting edge enters different tooth spaces of said member on successive strokes and successively cuts in each tooth space, changing the relative position of the cutting edge and member between cuts in the same tooth space until a; final position is reached, and describing and finish-cutting the entire engaged tooth side in a single stroke per tooth in said final position.

14. The method of cutting teeth having a varying dis-\ tance from the axis of a toothed member, which comprises rotating said member continuously in the same direction on its axis, moving a form-cutting edge about and along another axis in repetitive strokes in time with the turning motion of said member, to describe a helical path across the face of said member, whereby said edge enters different tooth spaces on successive strokes, changing the relative position of the cutting edge and member between successive cuts in the same tooth space until a final position is reached, and finishing each entire tooth side in a single stroke in said final position.

15. The method of cutting teeth having a varying distance from the axis of a toothed member, which comprises rotating said member continuously in the same direction on its axis, moving a tool containing a pair of cutting edges about and along another axis in repetitive strokes in time with the turning motion of said member across the face of said member, whereby said pair of cutting edges pass through adjacent tooth spaces in each stroke and each cutting edge enters a different tooth space of said member on successive strokes, changing the relative position of the tool and member between successive cuts of a cutting edge in the same tooth space until a final position is reached, and finishing each entire tooth side in a single stroke in said final position.

16. The method of cutting the sides of teeth whose two tooth ends have diiierent distances from the axis of a gear, which comprises moving a side-cutting edge about and along an axis across the face of said gear so that it describes in space a helical surface of constant lead, the points of said side-cutting edge having varying distances from said axis, said axis being inclined to the direction of the gear axis and its extension intersecting said gear inside of the reach of its teeth, effecting feeding motion between said cutting edge and gear, repeating said helical motion in difierent turning positions of said gear and in diflYerent relative feed positions in further cutting passes, to finally describe and finish-cut each entire tooth side in a single pass in the final feed position.

17. The method of cutting straight teeth on a bevel gear, which comprises positioning a pair of concavely curved form-cutting edges adjacent opposite sides of said teeth, the curvature of said cutting edges varying along their height, moving said cutting edges simultaneously across the face of said gear, said motion being about and along an axis on each of said cutting edges and the cutting edges being so shaped that each cutting edge describes an involute helicoid at least approximately, said axis being inclined to a plane tangent to the engaged tooth zone and ofiset from said tooth zone, the involute helical surfaces described by said two cutting edges being of opposite hand, effecting feeding motion between said cutting edges and said gear, and repeating said helical motions in turning positions of said gear and in difierent feed positions, to describe and out each entire tooth side in a single feed position.

References Cited in the file of this patent UNITED STATES PATENTS 1,610,995 Bostock et al. Dec. 14, 1926 1,673,488 Bishop June 12, 1928 1,870,325 Douglas Aug. 9, 1932 -2,038,897 Geny Apr. 28, 1936 2,324,182 Wildhaber July 13, 1943 2,775,922 Wildhaber Ian. 1, 1957 

